Casting simulation method

ABSTRACT

Provided is a casting simulation method capable of expressing influence of different inelastic strains produced at different temperatures on strain hardenability at room temperature. The following amount of effective equivalent inelastic strain ε effective inelastic  is substituted into a constitutive equation in which an amount of equivalent inelastic strain is used as a degree of work hardening: 
       an amount of effective equivalent inelastic strain ε effective inelastic =∫ o   t   {h   (T)   /h   (RT) }{(Δε inelastic   /Δt )} dt  
 
     , where T denotes a temperature with inelastic strain, h (T)  denotes an increment of yield strength at room temperature with respect to an amount of inelastic strain at the temperature with inelastic strain, h (RT)  denotes an increment of yield strength at room temperature with respect to an amount of inelastic strain applied at room temperature, h (T) /h (RT)  denotes an effective inelastic strain coefficient α(T), Δε inelastic /Δt denotes an equivalent inelastic strain rate, and t denotes a time from 0 second in analysis.

TECHNICAL FIELD

This disclosure relates to a casting simulation method using thermalstress and deformation analysis.

BACKGROUND

Conventionally, elasto-plastic constitutive equation is used in analysisprograms in order to predict the residual stress and deformation thatoccurs as metals processed at high temperature, such as cast products,hot forged products, and hot rolled products, cool back to roomtemperature. JP2007330977A (PTL 1) and Dong Shuxin, Yasushi Iwata,Toshio Sugiyama, and Hiroaki Iwahori, “Cold Crack Criterion for ADC12Aluminum Alloy Die Casting”, Casting Engineering, 81(5), 2009, pp.226-231, ADC12 (NPL 1) are reference examples of Toyota Central R&DLabs. Inc.

However, in elasto-plastic constitutive equation and elasto-viscoplasticconstitutive equation that do not take into account recovery, the amountof equivalent inelastic strain (including the amount of plastic strainand the amount of viscoplastic strain) is used as a measure ofhardening, and inelastic strain that occurs at such a high temperatureat which recovery occurs simultaneously with deformation is also treatedas contributing to work hardening as much as inelastic strain producedat room temperature. This causes an unrealistic increase in yield stressat room temperature, causing problems in the accuracy of thermal stressanalysis.

To address this issue, there have been proposed a number of constitutiveequations that can take into account some recovery. Any of theseconstitutive equations, however, fail to give physical or experimentalgrounds for how recovery should be considered, and thus suffer fromproblems with prediction accuracy.

The problems of conventional findings related to the above-describedinventions will be described below.

CITATION LIST Patent Literature

-   -   PTL 1: JP2007330977A    -   PTL 2: Japanese Patent Application No. 2014-004578

Non-Patent Literature

-   -   NPL 1: Dong Shuxin, Yasushi Iwata, Toshio Sugiyama, and Hiroaki        Iwahori, “Cold Crack Criterion for ADC12 Aluminum Alloy Die        Casting”, Casting Engineering, 81(5), 2009, pp. 226-231, ADC12    -   NPL 2: Hallvard G. Ejar and Asbjorn Mo, “ALSPEN-A mathematical        model for thermal stresses in direct chill casting of aluminum        billets”, Metallurgical Transactions B, December 1990, Volume        21, Issue 6, Pages 1049-1061    -   NPL 3: W. M. van Haaften, B. Magnin, W. H. Kool, and L.        Katgerman, “Constitutive behavior of as-cast AA1050, AA3104, and        AA5182”, Metallurgical and Materials Transactions A, July 2002,        Volume 33, Issue 7, Pages 1971-1980    -   NPL 4: Alankar Alankar and Mary A. Wells, “Constitutive behavior        of as-cast aluminum alloys AA3104, AA5182 and AA6111 at below        solidus temperatures”, Materials Science and Engineering A,        Volume 527, Issues 29-30, 15 Nov. 2010, Pages 7812-7820

SUMMARY Technical Problem

Hallvard et al. (NPL 2) proposes a constitutive equation that isexpressed as Constitutive Eq. (I) below representing the relationshipbetween stress and inelastic strain as described below, and thatconsiders inelastic strain produced at or above a certain temperature asnot contributing to hardening, while the other produced below thattemperature as contributing to work hardening. From a metallurgicalviewpoint, however, it is clear that recovery does not happen suddenlyat a certain temperature. Therefore, this constitutive equation has aproblem.

Constitutive Eq. (I) representing the relationship between stress andinelastic strain:

$\overset{\_}{\sigma} = {{f\left( {\alpha,{\overset{\_}{\overset{.}{ɛ}}}_{p},T} \right)} = {{c(T)}\left( {\alpha + \alpha_{0}} \right)^{n{(T)}}\left( {\overset{\_}{\overset{.}{ɛ}}}_{p} \right)^{m{(T)}}}}$${d\; \alpha} = \left\{ \begin{matrix}{d\; {\overset{\_}{ɛ}}_{p}} & {{{when}\mspace{14mu} T} \leq T_{0}} \\0 & {otherwise}\end{matrix} \right.$

Van Haaften et al. (NPL 3) proposes a constitutive equation that isexpressed as Constitutive Eq. (II) below between stress and inelasticstrain, a function a which is 0 at high temperature and 1 at lowtemperature is used to smoothly consider the contributions of inelasticstrain produced at different temperatures to work hardening.

However, the amount of cumulative inelastic strain is directlymultiplied by a, which is not in incremental form, and inelastic strainproduced at high temperature eventually contribute to work hardening atlow temperature. Thus, as in Constitutive Eq. (I), the constitutiveequation proposed by NPL 3 inevitably involves an unrealistic increasein yield stress.

Constitutive Eq. (II) representing the relationship between stress andinelastic strain:

$\overset{.}{ɛ} = {{A\left\lbrack {\sinh \left( \frac{\sigma_{ss}}{\sigma_{0}} \right)} \right\rbrack}^{nH}{\exp \left( \frac{- Q}{RT} \right)}}$$\sigma_{H} = {\left( {\sigma_{0} + {k\sqrt{{\alpha (T)}ɛ}}} \right) \cdot {f(Z)}}$${f(Z)} = {\min \left( {1,{\arcsin \mspace{14mu} {h\left( \frac{Z}{A} \right)}^{mH}}} \right)}$$\alpha = \frac{1}{1 + {\exp \left( {a_{0} + {a_{1}T}} \right)}}$

Moreover, Alankar et al. (NPL 4) proposes a constitutive equation thatis expressed as Constitutive Eq. (III) below representing therelationship between stress and inelastic strain. With this constitutiveequation, recovery occurs more frequently as the ratio of a workhardening index at high temperature n_((T)) to a work hardening index atroom temperature n_(RT) decreases. However, there are no metallurgicalgrounds for considering that the ratio n_((T))/n_(RT) determines theratio between plastic strain contributing to work hardening and creepstrain not contributing to work hardening (strain making no contributionto work hardening). Additionally, it is not specified how to identifymaterial constants in the constitutive equation when inelastic strain isdivided into creep strain and plastic strain. Thus, K_((T)), n_((T)),and m_((T)) in the following equation cannot be determined accurately.

Constitutive Eq. (III) representing the relationship between stress andinelastic strain:

$\sigma = {{K(T)}{ɛ_{p}^{n{(T)}}\left( \frac{{\overset{.}{ɛ}}_{p}}{ɛ_{0}} \right)}^{m{(T)}}}$$\sigma \; {K(T)}\left( {ɛ_{p} + ɛ_{p_{0}}} \right)^{n{(T)}}\left( {{\overset{.}{ɛ}}_{p} + {\overset{.}{ɛ}}_{p_{0}}} \right)^{m{(T)}}$$ɛ_{creep} = {ɛ_{plastic}\left( {1 - \frac{n_{(T)}}{n_{RT}}} \right)}$${\% \mspace{14mu} {strain}\mspace{14mu} {softening}} = {\left( {1 - \frac{n_{(T)}}{n_{RT}}} \right) \times 100}$

Recently, one of the applicants of the present application proposed aconstitutive equation to solve at least part of the above problems inJapanese Patent Application No. 2014-004578 (PTL 2, an unpublishedearlier application). This is expressed as a constitutive equationexpressing the relationship between stress and elastic strain/inelasticstrain as explained below. In this constitutive equation, inelasticstrain is divided into plastic strain contributing to work hardening andcreep strain not contributing to work hardening, and inelastic strainproduced at high temperature is expressed mainly by creep strain. Inthis case, measures are taken to prevent the yield stress fromexcessively rising at room temperature by causing plastic strain todevelop gradually as the temperature decreases.

Constitutive Eq. expressing the relationship between stress and elasticstrain/inelastic strain:

ɛ = ɛ_(elastic) + ɛ_(plastic) + ɛ_(creep) + ɛ_(thermal)σ = E ⋅ ɛ_(elastic)$ɛ_{creep} = {A \cdot {\exp \left( \frac{\sigma}{RT} \right)}}$σ = f(ɛ_(p), T),

where f denotes a yield function.

With the constitutive equation proposed by PTL 2, however, theproportion of plastic strain and creep strain is determined based on theshape of a stress-equivalent inelastic strain curve obtained atdifferent temperatures, rather than on the metallurgical grounds. Inaddition, as a steady-state creep law is used, it is inevitable toestimate plastic strain excessively beyond the actual value, whileestimating creep strain low in low strain regions. Therefore, problemsremain in the prediction accuracy of residual stress and deformation.

Recent metallurgical findings revealed that the extent to whichinelastic strain at high temperature contributes to work hardening atroom temperature depends on the composition of the alloy, thermalhistory such as in heat treatment, and solidified structure.

To date, however, there has been no such constitutive equation thattakes into account all the factors listed above in order to predict theresidual stress and deformation that occurs as metals processed at hightemperature, such as cast products, hot forged products, and hot rolledproducts, cool back to room temperature.

It is thus desirable at present in construction of a constitutiveequation to experimentally clarify how inelastic strain at hightemperature contributes to work hardening at room temperature, andreflect it in the constitutive equation.

However, none of the conventional constitutive equations can reflect“the influence of inelastic strain produced at different temperatures onwork hardening at room temperature” that is determined on anexperimental or theoretical basis. There has also been no finding thatshows how to determine material constants in a constitutive equationthat can reflect this effect.

Solution to Problem

The present disclosure has been developed in view of the abovecircumstances, and provides a casting simulation method using thermalstress and deformation analysis, in which an amount of equivalentinelastic strain effective for work hardening is determined bymultiplying an equivalent inelastic strain rate calculated in theanalysis by an effective inelastic strain coefficient α representing aproportion of inelastic strain contributing to work hardening to obtainan effective inelastic strain rate, and integrating it over a time from0 second in analysis, and the amount of equivalent inelastic strain thusobtained is used as a measure of work hardening in a constitutiveequation.

To solve the above-described problems, the present disclosure alsoprovides a method of experimentally determining an effective inelasticstrain coefficient α(T) which represents the contributions of inelasticstrain produced at different temperatures to work hardening.

Specifically, the primary features of this disclosure are as describedbelow.

1. A casting simulation method capable of expressing influence ofdifferent inelastic strains produced at different temperatures on workhardening, namely on increase in yield stress, at room temperature, theinfluence varying with differences in recovery at the differenttemperatures, by introducing an amount of effective equivalent inelasticstrain to an elasto-plastic constitutive equation and/or anelasto-viscoplastic constitutive equation in which an amount ofequivalent inelastic strain is used as a degree of work hardening,namely an amount of increase in yield stress, such as an elasto-plasticconstitutive equation in which a yield function is expressed asf=f(σ_(eff),ε_(eff) ^(p),T) or an elasto-viscoplastic constitutiveequation in which a relation between equivalent stress, equivalentviscoplastic strain, viscoplastic strain rate, and temperature isexpressed as σ_(eff.)=F(ε_(eff.) ^(vp),{dot over (ε)}_(eff.) ^(vp),T),wherein an amount of effective equivalent inelastic strainε_(effective inelastic) obtained by Eq. (1) below is used:

the amount of effective equivalent inelastic strainε_(effective inelastic)=∫_(o) ^(t) {t _((T)) /h _((RT))}{(Δε_(inelastic)/Δt)}dt   (1)

, where T denotes a temperature with inelastic strain, h_((T)) denotesan increment of yield strength at room temperature with respect to anamount of inelastic strain at the temperature with inelastic strain,h_((RT)) denotes an increment of yield strength at room temperature withrespect to an amount of inelastic strain applied at room temperature,h_((T))/h_((RT)) denotes an effective inelastic strain coefficient α(T),Δε_(inelastic)/Δt denotes an equivalent inelastic strain rate, and tdenotes a time from 0 second in analysis.

2. The casting simulation method according to 1., the effectiveinelastic strain coefficient α(T) is obtained by: applying differentinelastic pre-strains to a test piece at different temperatures; coolingthe test piece to room temperature; performing a tensile test or acompression test on the test piece at room temperature; and measuringinfluence of amounts of the inelastic pre-strains applied at thedifferent temperatures on the increase in yield stress.

3. The casting simulation method according to 1. or 2., wherein astress-equivalent inelastic strain curve is transformed into astress-effective equivalent inelastic strain curve using α(T), and basedon the stress-effective equivalent inelastic strain curve, adetermination is made of a material constant in a constitutive equationto which the amount of effective equivalent inelastic strainε_(effective inelastic) is introduced.

4. The casting simulation method according to any one of 1. to 3.,wherein when α(T) is substituted into Eq. (1) in a temperature range inwhich α(T) is 0 or a negative value and a stress-equivalent inelasticstrain curve at the temperature T indicates work hardening, α(T) iscorrected from 0 or the negative value to a small positive value.

Advantageous Effect

According to the present disclosure, it is possible to eliminate thephysical and metallurgical irrationality of the conventional methods,and to express, based on the experimental fact, the influence ofdifferent inelastic strains produced at different temperatures on workhardening at room temperature, which influence varies with differencesin recovery at the different temperatures. As a result, it is possibleto more accurately simulate the residual stress and deformationoccurring as metals processed at high temperature cool back to roomtemperature.

BRIEF DESCRIPTION OF THE DRAWING

In the accompanying drawings:

FIG. 1 a graph conceptually illustrating the temperature history of atest piece in a test for determining an effective inelastic straincoefficient;

FIG. 2 is a graph conceptually illustrating the influence of amounts ofinelastic pre-strains at different temperatures on yield stress rise atroom temperature;

FIG. 3 is a graph illustrating a temperature history of JIS ADC12 usedin tests for determining an effective inelastic strain coefficient inexamples;

FIG. 4 is a graph illustrating the experimental results obtained inexamples of examining the influence of inelastic pre-strains atdifferent temperatures on the 0.2% offset yield stress of JIS ADC12 atroom temperature;

FIG. 5 is a graph illustrating material constants K, m, n, and αobtained for JIS ADC12 in examples;

FIG. 6 is a graph illustrating the results obtained in examples ofcomparing experimental values with calculated values according to thepresent disclosure in stress-strain curves for JIS ADC12 at roomtemperature;

FIG. 7 is a graph illustrating the results obtained in examples ofcomparing experimental values with calculated values according to thepresent disclosure in stress-strain curves for JIS ADC12 at 200° C.;

FIG. 8 is a graph illustrating the results obtained in examples ofcomparing experimental values and calculated values according to thepresent disclosure in stress-strain curves for JIS ADC12 at 250° C.;

FIG. 9 is a graph illustrating the results obtained in examples ofcomparing experimental values and calculated values according to thepresent disclosure in stress-strain curves for JIS ADC12 at 300° C.;

FIG. 10 is a graph illustrating the results obtained in examples ofcomparing experimental values and calculated values according to thepresent disclosure in stress-strain curves for JIS ADC12 at 350° C.;

FIG. 11 is a graph illustrating the results obtained in examples ofcomparing experimental values and calculated values according to thepresent disclosure in stress-strain curves for JIS ADC12 at 400° C.;

FIG. 12 is a graph illustrating the results obtained in examples ofcomparing experimental values and calculated values according to thepresent disclosure in stress-strain curves for JIS ADC12 at 450° C.;

FIG. 13 is a diagram illustrating “the influence of inelasticpre-strains at room temperature and at 450° C. on the increase in yieldstress at room temperature” calculated in examples using theconventional extended Ludwik's law;

FIG. 14 is a graph illustrating “the influence of inelastic pre-strainsapplied at different temperatures on increase in yield stress at roomtemperature” calculated in examples according to the present disclosure;

FIG. 15 is a graph illustrating the results obtained in examples ofcomparing experimental values and calculated values according to thepresent disclosure to examine the influence of inelastic pre-strainsapplied at different temperatures on the increase in 0.2% offset yieldstress at room temperature; and

FIG. 16 is a graph illustrating experimental results in examples ofexamining the influence of inelastic pre-strains applied at differenttemperatures on the 0.2% offset yield stress of FCD400 at roomtemperature.

DETAILED DESCRIPTION

The following describes the present disclosure in detail.

As used herein, an equivalent inelastic strain rate calculated bythermal stress analysis is multiplied by an “effective inelastic straincoefficient α(T) indicative of temperature dependency”, which ispreferably experimentally determined, and the result is used as aneffective inelastic strain rate. The effective inelastic strain rate isthen integrated over a time from 0 second in analysis to determine theamount of effective equivalent inelastic strain, which in turn is, inplace of the amount of equivalent inelastic strain, used as a measure ofhardening in a constitutive equation.

The amount of effective equivalent inelastic strain is applicable to anyconstitutive equation, whether an elasto-plastic constitutive equationor an elasto-viscoplastic constitutive equation, as long as it is aconstitutive equation using the amount of equivalent inelastic strainconventionally as a measure of work hardening as described in paragraph0008.

Specific examples are an elasto-plastic constitutive equation and/or anelasto-viscoplastic constitutive equation in which the amount ofequivalent inelastic strain is used as a degree of work hardening(namely an amount of increase in yield stress) such as an elasto-plasticconstitutive equation in which a yield function is expressed asf=f(σ_(eff),ε_(eff) ^(p),T) or an elasto-viscoplastic constitutiveequation in which a relation between equivalent stress, equivalentviscoplastic strain, viscoplastic strain rate, and temperature isexpressed as σ_(eff.)=F(ε_(eff.) ^(vp),{dot over (ε)}_(eff.) ^(vp),T).

The determination of the effective inelastic strain coefficient α(T) iscarried out by applying inelastic pre-strains of different magnitudes atthe corresponding temperatures at which α(T) is to be obtained, thencooling to room temperature, conducting a tensile test or a compressiontest, and measuring the increase in yield stress.

The following describes how to determine α(T) in detail.

A tensile test piece is heated in a way as presented in the temperaturehistory in FIG. 1. For example, let T₃ be the temperature at which α(T)is to be determined, a single inelastic pre-strain is applied at T₃ inT₁, T₂, T₃, . . . , T_(n) per test. Then, the test piece is cooled toroom temperature and subjected to a tensile test or a compression testat room temperature. Then, measurement is made of the increase in yieldstress. This procedure is repeated for inelastic pre-strains atdifferent temperatures and at different magnitudes. In this way, theinfluence of the amounts of the inelastic pre-strains applied at thedifferent temperatures on the increase in yield stress at roomtemperature is measured.

Desirably, the same temperature history is set for all test conditions.The reason for this is to eliminate the influence of the temperaturehistory of the test piece on the measured values.

In the context of the present disclosure, the above-described test isnot limited to a particular test, and may be a tensile test or acompression test as long as it can provide a stress-strain curve andenables measurement of yield stress. For the tensile test in thisdisclosure, for example, a publicly-known and widely-used tensile testmay be used, such as JIS Z 2241:2011. For the compression test, forexample, a publicly-known and widely-used compression test may be used,such as JIS K 7181:2011.

The results obtained in the above test are conceptually illustrated inFIG. 2. FIG. 2 illustrates the influence of inelastic pre-strainsapplied at different temperatures on the increase in yield stress atroom temperature. As can be seen from FIG. 2, if a gradient of yieldstress with respect to the amount of inelastic pre-strains applied atroom temperature is expressed as h_((RT)), a gradient of yield stresswith respect to the amount of inelastic pre-strains applied at differenttemperatures can be similarly expressed as h_((T)). Thus, at eachtemperature T, the value of h_((T))/h_((RT)) is used as an effectiveinelastic strain coefficient α(T), which is indicative of how much theinelastic strain produced at the temperature T contribute to workhardening at room temperature with respect to the inelastic strainproduced at room temperature at which all the inelastic strains producedcontribute to work hardening.

In a temperature range in which an effective inelastic straincoefficient α(T) is experimentally determined to be 0 or a negativevalue, i.e., in which inelastic strain applied at the temperature Tshould not contribute to work hardening at room temperature, if thestress-equivalent inelastic strain curve at the temperature T indicateswork hardening and if the effective inelastic strain coefficient α(T) is0, then a constitutive equation to which the amount of effectiveequivalent inelastic strain is introduced involves no effectiveinelastic strain, and thus is not able to express work hardening inprinciple. In other words, the stress-strain curve displayselasto-perfectly plastic behavior or elasto-perfectly viscoplasticbehavior. In that case, the reproducibility of the stress-strain curvedeteriorates, resulting in lower accuracy of predictions on thermalstress and deformation.

An effective inelastic strain is produced as long as the effectiveinelastic strain coefficient α(T) is not 0, making it possible toexpress work hardening in the stress-inelastic strain curve at thetemperature T. Accordingly, even in a temperature range with theinelastic strain coefficient α(T) being 0 or a negative value, if apositive small value, rather than 0 or a negative value, is correctedappropriately for use as an effective inelastic strain coefficient α(T),it is possible to express work hardening in a stress-equivalentinelastic strain curve with a slight ineffective inelastic strain. Thevalue of α(T) at the time of correction is in a range of 0<α<0.5, anddesirably 0<α<0.1, although it depends on the alloy type. As an example,α(T) is corrected by the following linear interpolation using aneffective inelastic strain coefficient at maximum temperature α_(min) ina temperature range in which α(T) is experimentally determined to benon-zero and a maximum temperature T_(max) (which may alternatively be asolidus temperature) in a temperature range in which α is experimentallydetermined to be 0:

$\begin{matrix}{{\alpha_{extrapolation}(T)} = {{\frac{0 - \alpha_{\min}}{T_{\max} - T_{\min}}\left( {T - T_{\min}} \right)} + \alpha_{\min}}} & \left( {T > T_{\min}} \right)\end{matrix},$

where

α_(extrapolation)(T) denotes a value of a as corrected in a temperaturerange in which α is experimentally determined to be 0;

T_(max) denotes the maximum temperature (which may alternatively be asolidus temperature) in a temperature range where a is experimentallydetermined to be 0,

T_(min) denotes a maximum temperature in a temperature range in which αis experimentally determined to be non-zero;

α_(min) denotes a value of α in T_(min); and

T denotes a temperature above T_(min)

As an example, material constants in a constitutive equation to whichthe amount of effective inelastic strain is introduced are determined asexplained below in the case of introducing the amount of effectiveinelastic strain to the constants (K(T), n(T), and m(T)) of the extendedLudwik's law.

The extended Ludwik's law is as follows. ε₀ is a constant necessary forcalculation and usually a small value of 1×10⁻⁶.

σ=K(T)(ε_(inelastic)+ε₀)^(n(T))({dot over (ε)}_(inelastic))^(m(T))

When introducing the amount of effective inelastic strain, it isexpressed as:

σ=K(T)(ε_(effective inelastic)+ε₀)^(n(T))({dot over(ε)}_(inelastic))^(m(T))

The term with an index n_((T)) representing the degree of work hardeningincludes the amount of effective equivalent inelastic strain as avariable. In addition, the term with an index m_((T)) representing thestrain rate dependence of the stress-strain curve includes an equivalentinelastic strain rate as a variable. Since the equation as a wholeincludes the amount of effective equivalent inelastic strain, for eachtemperature, by substituting the inelastic strain rate into the termwith m_((T)), K_((T)), n_((T)), and m_((T)) are determined by numericaloptimization to fit the stress-effective equivalent inelastic straincurve.

Following the above procedure, the amount of effective equivalentinelastic strain ε_(effective inelastic) is determined. In thedisclosure, the amount of effective equivalent inelastic strain is usedto simulate the influence of inelastic strain applied at the temperaturein question on work hardening at room temperature. Although details ofthe procedures for casting simulation will be described in the Examples,the points are summarized as follows.

In the simulation according to the disclosure, it is possible to adopt aconstitutional expression that uses the amount of equivalent inelasticstrain as a measure of work hardening conventionally used for castingsimulation. An exemplary elasto-plastic constitutive equation is:

ε_(ij)=ε^(e) _(ij)+ε^(p) _(ij)

σ_(ij) =D _(ijkl)(T)ε^(e) _(kl)

f=f(σ_(eff),ε_(eff) ^(p) ,T)

, where T is the temperature, σ_(ij) is the stress, σ_(eff.) is theequivalent stress, ε_(ij) is the total strain, ε^(e) _(ij) is theelastic strain, ε^(p) _(ij) is the plastic strain, f is the yieldfunction, and D_(ijkl) is the fourth-order constitutive tensor.

Alternatively, an exemplary elasto-viscoplastic constitutive equationis:

ε_(ij)=ε^(e) _(ij)+ε^(vp) _(ij)

σ_(ij) =D _(ijkl)(T)ε^(e) _(kl)

σ_(eff.) =F(ε_(eff.) ^(vp),{dot over (ε)}_(eff.) ^(vp) ,T)

, where σ_(eff.) is the equivalent stress, ε_(eff.) ^(vp) is theequivalent viscoplastic strain, {dot over (ε)}_(eff.) ^(vp) is theequivalent viscoplastic strain rate.

To these constitutive equations, the amount of effective equivalentinelastic strain ε_(effective inelastic) defined by Eq. (1) according tothe present disclosure, instead of the amount of equivalent inelasticstrain conventionally used, may be introduced or substituted.

The well-known and widely-used procedures for casting simulation are:

(I) element creation step;

(II) element definition step;

(III) heat transfer analysis step;

(IV) thermal stress analysis step; and

(V) analysis result evaluation step.

In the present disclosure, material constants in a constitutive equationare determined in step (II) using an equivalent stress-effectiveequivalent inelastic strain curve, and are input to a constitutiveequation to which the amount of effective inelastic strain isintroduced. Then, in the thermal stress analysis step (IV), the amountof effective equivalent inelastic strain is calculated, and the result,instead of the amount of equivalent inelastic strain conventionallyused, is used as a parameter representing the amount of work hardeningto calculate thermal stress.

EXAMPLES

The following describes how the present disclosure enables predictionwith high accuracy of the influence of amounts of inelastic pre-strainsproduced at different temperatures on the work hardening behavior atroom temperature in a typical aluminum die-casting alloy, JIS ADC12,using an extended Ludwik equation, which is a typicalelasto-viscoplastic constitutive equation.

A specific form of the equation before and after the introduction of theamount of effective inelastic strain is as presented in paragraph 0032.

Firstly, a typical aluminum die-casting alloy, JIS ADC12, is analyzedfor an effective inelastic strain coefficient α(T) and materialconstants (K(T), n(T), and m(T)) at each temperature according to theprocedures described in paragraphs 0026 to 0033.

Firstly, tensile tests were performed to obtain stress-equivalentinelastic strain curves required to determine material constants K(T),n(T), and m(T). Stress-inelastic strain curves were obtained under a setof conditions including: experimental strain rates of 10⁻³/s and 10⁻⁴/sand test temperatures of RT, 200° C., 250° C., 300° C., 350° C., 400°C., and 450° C. Each test pieces was obtained by casting JIS ADC12 in acopper mold and processing it into the shape of a tensile test piece.

In these tests, all the test pieces were heated from room temperature to450° C., then subjected to heat treatment at 450° C. for 1 hour to causeprecipitates to be re-dissolved, and cooled to the test temperature assoon as possible so that the mechanical properties at the time ofcooling can be examined accurately. As soon as the test temperature wasreached, the tensile test was carried out.

In addition, tests for determining an effective inelastic straincoefficient α(T) were carried out at RT, 200° C., 250° C., 300° C., 350°C., 400° C., and 450° C. After solution treatment at 450° C. for 1 hour,the temperature was lowered to a temperature at which the targetinelastic pre-strain was to be applied to the test piece following thetemperature history presented in FIG. 3, and after the targettemperature was reached, an inelastic pre-strain was applied in tensionwhile retaining the test piece at the target temperature. At eachtemperature, two or three different pre-strains were applied.

After application of pre-strains, each test piece was cooled to roomtemperature and quenched with dry ice to eliminate the effect of theincrease in yield stress caused by natural aging. Then, the 0.2% offsetyield stress was determined by conducting a tensile test on each testpiece at room temperature. The results are presented in FIG. 4. For atest piece not imparted with inelastic pre-strain, the 0.2% offset yieldstress was 106 MPa. Based on the results presented in the figure,determinations of an effective inelastic strain coefficient α(T) weremade of the rate of increase in 0.2% offset yield stress with respect tothe increase in inelastic pre-strain at each temperature, that is,h_((T)), and of the value of h_((T)) at room temperature. The resultsare presented in FIG. 5. In the temperature range from 350° C. to 400°C., the effective inelastic strain coefficient is 0 by definition, yetany inelastic strain produced contributes to work hardening at roomtemperature, although not depending on the quantity, and exhibits workhardening even in a stress-equivalent inelastic strain curve. At 450°C., the effective inelastic strain coefficient becomes 0 and anyinelastic strain produced does not contribute to work hardening at roomtemperature. From this experimental fact, as explained in paragraph0031, α was set to 0.000185 at 350° C., 0.0000927 at 400° C., and 0 at450° C., while correcting the effective inelastic strain coefficient at350° C. and 400° C. from 0 to a very small positive value, so that workhardening could be expressed in the stress-inelastic strain curve in thetemperature range of 350° C. to 400° C. and almost no effectiveinelastic strain would be produced.

By using the effective inelastic strain coefficient α_((T)) thusobtained, the stress-equivalent inelastic strain curve obtained inparagraph 0040 was transformed into a stress-effective equivalentinelastic strain curve, and the values of K_((T)), m_((T)), and n_((T))were obtained as described in paragraphs 0032 and 0033. The values ofK_((T)), m_((T)), and n_((T)) are presented in FIG. 5.

FIGS. 6 to 12 each illustrate stress-inelastic strain curves to compareexperimental values with calculated values according to the extendedLudwik equation to which the amount of effective equivalent inelasticstrain determined by Eq. (1) is introduced. In each case, tensile testswere performed at strain rates of 10⁻³/s and 10⁻⁴/s at RT, 200° C., 250°C., 300° C., 350° C., 400° C., and 450° C. Except for 450° C., all theconstitutive equations incorporating the amount of effective equivalentinelastic strain accurately reproduced the strain rate dependence andthe shape of the corresponding stress-inelastic strain curve. In thiscase, we faithfully followed the data obtained in the limitedexperimental temperature range and considered 450° C. as T_(max)described in paragraph 0031. Thus, as described in paragraphs 0030 and0031, the stress-inelastic strain curve displays elasto-perfectlyviscoplastic behavior at 450° C. Therefore, there is a divergencebetween the experimental values and the calculated values.

It is conceivable, however, that if α is corrected as described inparagraph 0031 with the temperature of α=0, T_(max) in paragraph 0031,being temporarily set as a liquidus temperature, the reproducibility ofthe stress-inelastic strain curve also improves.

For extended Ludwik equations incorporating or not incorporating (in thecase of a conventional example) the amount of effective equivalentinelastic strain according to the present disclosure, calculation wasmade to determine the effect of inelastic pre-strains applied atdifferent temperatures on the yield stress at room temperature, and thecalculation results were compared as presented in FIG. 13 (aconventional example) and FIG. 14 (an example of the presentdisclosure).

In the conventional extended Ludwik equation not incorporating theamount of effective equivalent inelastic strain, in principle, inelasticstrains produced in different temperature ranges are all considered asequivalent to one another and included as a measure of hardening.Accordingly, as is clear from FIG. 13, inelastic strains produced at450° C. as well as those produced at room temperature contributed to anincrease in yield stress, and thus to an unrealistic increase in yieldstress. It is noted here that inelastic strains applied at othertemperatures also have the same results as at 450° C. with overlappingplots, and thus they are omitted in the figure.

On the other hand, as can be seen from FIG. 14, in the expanded Ludwikequation incorporating the amount of effective inelastic strainaccording to the present disclosure, the influence of the amount ofinelastic pre-strain on work hardening gradually increased withdecreasing temperature, which fact reproduced the behavior observed inthe experimental results.

To examine the influence of inelastic pre-strain at differenttemperatures on the yield stress at room temperature, a comparison wasmade between experimental values and calculated values according to theextended Ludwik equation to which the amount of effective equivalentinelastic strain is introduced, and the results are presented in FIG.15.

It can be seen from the figure that the analysis program incorporatingthe amount of effective equivalent inelastic strain could reproduce thebehavior at 300° C. or higher at which an increase in yield stress isindependent of the amount of inelastic pre-strain. In addition, thisprogram could accurately reproduce the behavior even at 300° C. or lowerat which an increase in yield stress depends on the amount of inelasticpre-strain.

FIG. 16 depicts the influence of experimentally obtained inelasticstrain at high temperature on the 0.2% offset yield stress of a typicalcast iron, JIS FCD400, at room temperature.

It can be seen from the figure that pre-strain applied at 700° C. doesnot contribute to work hardening at room temperature. In contrast,inelastic strain applied at 350° C. contributes to work hardening atroom temperature and the amount of work hardening is proportional to theamount of inelastic strain applied. This behavior is identical to thatobserved in ADC12, and from this follows that the present disclosure isalso applicable to FCD400.

1. A casting simulation method capable of expressing influence ofdifferent inelastic strains produced at different temperatures on workhardening, namely on increase in yield stress, at room temperature, theinfluence varying with differences in recovery at the differenttemperatures, by introducing an amount of effective equivalent inelasticstrain to an elasto-plastic constitutive equation and/or anelasto-viscoplastic constitutive equation in which an amount ofequivalent inelastic strain is used as a degree of work hardening,namely an amount of increase in yield stress, such as an elasto-plasticconstitutive equation in which a yield function is expressed asf=f(σ_(eff),ε_(eff) ^(p),T) or an elasto-viscoplastic constitutiveequation in which a relation between equivalent stress, equivalentviscoplastic strain, viscoplastic strain rate, and temperature isexpressed as σ_(eff.)=F(ε_(eff.) ^(vp),{dot over (ε)}_(eff.) ^(vp),T),wherein an amount of effective equivalent inelastic strainε_(effective inelastic) obtained by Eq. (1) below is used:the amount of effective equivalent inelastic strainε_(effective inelastic)=∫_(o) ^(t) {h _((T)) /h _((RT))}{(Δε_(inelastic)/Δt)}dt   (1) , where T denotes a temperature with inelastic strain,h_((T)) denotes an increment of yield strength at room temperature withrespect to an amount of inelastic strain at the temperature withinelastic strain, h_((RT)) denotes an increment of yield strength atroom temperature with respect to an amount of inelastic strain appliedat room temperature, h_((T))/h_((RT)) denotes an effective inelasticstrain coefficient α(T), Δε_(inelastic)/Δt denotes an equivalentinelastic strain rate, and t denotes a time from 0 second in analysis.2. The casting simulation method according to claim 1, the effectiveinelastic strain coefficient α(T) is obtained by: applying differentinelastic pre-strains to a test piece at different temperatures; coolingthe test piece to room temperature; performing a tensile test or acompression test on the test piece at room temperature; and measuringinfluence of amounts of the inelastic pre-strains applied at thedifferent temperatures on the increase in yield stress.
 3. The castingsimulation method according to claim 1, wherein a stress-equivalentinelastic strain curve is transformed into a stress-effective equivalentinelastic strain curve using α(T), and based on the stress-effectiveequivalent inelastic strain curve, a determination is made of a materialconstant in a constitutive equation to which the amount of effectiveequivalent inelastic strain ε_(effective inelastic) is introduced. 4.The casting simulation method according to claim 1, wherein when α(T) issubstituted into Eq. (1) in a temperature range in which α(T) is 0 or anegative value and a stress-equivalent inelastic strain curve at thetemperature T indicates work hardening, α(T) is corrected from 0 or thenegative value to a small positive value.
 5. The casting simulationmethod according to claim 2, wherein a stress-equivalent inelasticstrain curve is transformed into a stress-effective equivalent inelasticstrain curve using α(T), and based on the stress-effective equivalentinelastic strain curve, a determination is made of a material constantin a constitutive equation to which the amount of effective equivalentinelastic strain ε_(effective inelastic) is introduced.
 6. The castingsimulation method according to claim 2, wherein when α(T) is substitutedinto Eq. (1) in a temperature range in which α(T) is 0 or a negativevalue and a stress-equivalent inelastic strain curve at the temperatureT indicates work hardening, α(T) is corrected from 0 or the negativevalue to a small positive value.
 7. The casting simulation methodaccording to claim 3, wherein when α(T) is substituted into Eq. (1) in atemperature range in which α(T) is 0 or a negative value and astress-equivalent inelastic strain curve at the temperature T indicateswork hardening, α(T) is corrected from 0 or the negative value to asmall positive value.
 8. The casting simulation method according toclaim 5, wherein when α(T) is substituted into Eq. (1) in a temperaturerange in which α(T) is 0 or a negative value and a stress-equivalentinelastic strain curve at the temperature T indicates work hardening,α(T) is corrected from 0 or the negative value to a small positivevalue.